Simplifying, Distributing, and Factoring - GED Math

Group all like terms by their order:

Simplify:

where

The numbers and fit those criteria. Therefore,

You can double check the answer using the FOIL method

Perhaps the easiest way to identify the factor is to take advantage of the factor theorem, which states that is a factor of polynomial if and only if . We substitute 1, 2, 4, and 9 for in the polynomial to identify the factor.

:

:

:

:

Only makes the polynomial equal to 0, so among the choices, only is a factor.

Perhaps the easiest way to identify the factor is to take advantage of the factor theorem, which states that is a factor of polynomial if and only if . We substitute and for in the polynomial to identify the factor.

:

:

:

:

Only makes the polynomial equal to 0, so of the four choices, only is a factor of the polynomial.

Factor all the way to its prime factorization.

can be factored as the difference of two perfect square terms as follows:

is a factor, and, as the sum of squares, it is a prime. is also a factor, but it is not a prime factor - it can be factored as the difference of two perfect square terms. We continue:

Therefore, all of the given polynomials are factors of , but is the correct choice, as it is not a prime factor.

is the sum of two cubes:

As such, it can be factored using the pattern

where ;

The first factor,as the sum of squares, is a prime.

We try to factor the second by noting that it is "quadratic-style" based on . and can be written as

;

we seek to factor it as

We want a pair of integers whose product is 1 and whose sum is . These integers do not exist, so is a prime.

is the prime factorization and the correct response is .

This can be most easily solved by first substituting for , and, subsequently, for :

This becomes quadratic in the new variable, and can be factored as

,

filling out the blanks with two numbers whose sum is and whose product is . Through some trial and error, the numbers can be seen to be .

Therefore, after factoring and substituting back,

The first factor, the sum of squares, is prime. The second factors as the difference of squares, so the final factorization is

.

Of the choices given, is correct.

can be seen to fit the pattern

:

where

can be factored as , so

.

does not fit into any factorization pattern, so it is prime, and the above is the complete factorization of the polynomial. Therefore, is the correct choice.

The area of a triangle is:

Use the FOIL Method to simplify.

To convert a base ten number to base five, divide the number by five, with the remainder being the digit in the units place; continue, dividing each successive quotient by five and putting the remainder in the next position to the left until the final quotient is less than five.

- 1 is the last digit.

- 2 is the second-to-last digit.

- 1 is the third-to-last digit; 2 is the first digit.

286 is equal to .

A number increased by 20% is equivalent to 100% of the number plus 20% of the number. This is taking 120% of the number, or, equivalently, multiplying it by 1.2.

Therefore, increased by 20% is 1.2 times this, or

.

Divide termwise:

This product fits the difference of cubes pattern, where :

so

This can be solved using the FOIL method. The steps are shown below.

Therefore, after reordering, the answer is:

A number decreased by 40% is equivalent to 100% of the number minus 40% of the number. This is taking 60% of the number, or, equivalently, multiplying it by 0.6.

Therefore, decreased by 40% is 0.6 times this, or

.

Raise a fraction to a negative power by raising its reciprocal to the power of the absolute value of the exponent. Then apply the power of a quotient rule:

Use the grouping technique, then distribute out the greatest common factor of each group as follows:

The common factor of the terms and can be found as follows:

, and the lesser of the two powers of is ; therefore, , their product. Distribute this out:

This is as far as we can go with the factoring.